# Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field

Shuichi Otake

研究成果: Article

### 抄録

In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to Sp or Ap.

元の言語 English 429-441 13 Linear and Multilinear Algebra 61 4 https://doi.org/10.1080/03081087.2012.689983 Published - 2013 4

### Fingerprint

Irreducible polynomial
Real Roots
Galois group
Number field
Counting
Distinct
Polynomial
Count
Isomorphic
Roots
Algebra
Form

### ASJC Scopus subject areas

• Algebra and Number Theory

### これを引用

：: Linear and Multilinear Algebra, 巻 61, 番号 4, 04.2013, p. 429-441.

研究成果: Article

@article{be2bdce719404892b64de6ca67be3114,
title = "Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field",
abstract = "In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to Sp or Ap.",
keywords = "Bezoutian, Galois group, irreducible polynomials, number of real roots",
author = "Shuichi Otake",
year = "2013",
month = "4",
doi = "10.1080/03081087.2012.689983",
language = "English",
volume = "61",
pages = "429--441",
journal = "Linear and Multilinear Algebra",
issn = "0308-1087",
publisher = "Taylor and Francis Ltd.",
number = "4",

}

TY - JOUR

T1 - Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field

AU - Otake, Shuichi

PY - 2013/4

Y1 - 2013/4

N2 - In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to Sp or Ap.

AB - In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to Sp or Ap.

KW - Bezoutian

KW - Galois group

KW - irreducible polynomials

KW - number of real roots

UR - http://www.scopus.com/inward/record.url?scp=84871881571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871881571&partnerID=8YFLogxK

U2 - 10.1080/03081087.2012.689983

DO - 10.1080/03081087.2012.689983

M3 - Article

AN - SCOPUS:84871881571

VL - 61

SP - 429

EP - 441

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 4

ER -